Cracking in Quasi-Brittle Materials Using Isotropic Damage Mechanics Tobias Gasch, PhD Student Co-author: Prof. Anders Ansell Comsol Conference 2016 Munich 2016-10-12
Contents Introduction Isotropic damage mechanics and localization Implementation in Comsol Multiphysics Examples Conclusions
Introduction What is a quasi-brittle material? Strain softening Fracture process zone (FPZ) Strong deterministic size effect All models presented are applicable to such materials but the presentation will focus on concrete Other examples are rocks, ceramics, ice Why analyze cracking of concrete? Failure Performance Durability
Contents Introduction Isotropic damage mechanics and localization Implementation in Comsol Multiphysics Examples Conclusions
Isotropic Damage Mechanics Progressive loss of material integrity due to propagation of material defects For example voids, cracks Leads to a degradation of the macroscopic stiffness Non-linear response concrete = cement matrix + aggregates The intact material carries a stress σ", often called the effective stress Over a unit volume of material the stress is then: σ = 1 ω σ" where 1 ω describes the relative amount of intact material, i.e. 0 ω 1 In a general formulation of damage mechanics, the scalar 1 ω is replaced by a 4 th order tensor
Isotropic Damage Mechanics Quasi-static formulation of the momentum balance and small strain kinematics:, σ + F / = 0; ε = 2 3 u + u5 ; + B.C. Assuming that the intact material is linear elastic, the constitute equation is given as: σ = 1 ω σ" = 1 ω C 78 : ε The non-linear response of the material is thus given by the evolution of the damage parameter ω
Isotropic Damage Mechanics A formulation following the framework by Oliver et al. (1990) Loading function f with the internal variable κ: f ε, κ ε ε + κ 0 The elastic domain is controlled by the equivalent strain ε : ε ε = 1 E max DE2,3,F C 78: ε D = 1 E max DE2,3,F σ D Loading/unloading conditions on the Kuhn-Tucker form: f 0, κ 0, κ f = 0 Damage evolution law ω κ for exponential softening: ω κ = 1 ε J κ exp κ ε J ε M In the Paper an additional defintion of ε is given and also another damage evolution law ω κ
Lack of mesh objectivity The stress-strain formulation leads to a strong mesh dependency of results No converging result upon mesh refinement Strains will localize in the narrowest possible region, i.e. a single element The amount of energy dissipated decrease with the element size Eventually the response becomes unstable Example from Jirásek (2011) Constitutive behavior Strain profile Force-displacement response More information is needed about the material and/or fracture process in needed!!!
Local formulation Crack-band method by Bažant and Oh (1983) At each material point (Gauss point): Supply information about the simulated FPZ, the crack-band width h O Construct unique stress-strain law from a stress-crack opening law given by G Q ε M = G Q /(f T h O ) + ε J /2 How to find an appropriate value of h O? Depends on for example interpolation order, element size and shape and the stress sate Here a projection method is used as proposed by Cervenka et al. (1990)
Non-local formulation Supply information about the material structure Width of the experimentally observed FPZ Non-local continuum that averages some variable/s over its spatial neighborhood Following Peerlings et al. (1996), higher order gradients are introduced in the constitutive law Non-local equivalent strain ε calculated as: ε c 3 ε = ε with the B.C. ε, n = 0 which replaces its local counterpart in the loading function Parameter c can be related to the width of the FPZ
Contents Introduction Isotropic damage mechanics and localization Implementation in Comsol Multiphysics Examples Conclusions
Implementation in Comsol Multiphysics Implemented in Comsol v5.2 (v5.2a) Utilizes the Linear Elastic material model of the Solid Mechanics interface, but: Introduces a new stress dmg.slxx (σ) which replaces the default stress solid.slxx (σ") in the weak expression This new stress is defined using equation-based-modelling: Domain ODE with the internal variable at the previously converged step κ Z8[ as dependent variable Discretized using Gauss point data shape functions Previous solution node The current state of damage calculated as κ = max(ε, κ \]^ ) σ" = σ 1 ω
Implementation in Comsol Multiphysics To calculate the crack-band width using the projection method the atlocal operator is used to obtain information about element coordinates and stress states The non-local model introduces an additional PDE to be solved with the non-local equivalent strain ε as dependent variable. - Discretized using Lagrange shape functions Only major difference is in the variable definition of κ Local Non-local κ = max(ε, κ \]^ ) κ = max(ε, κ \]^ )
Custom physics interface Created using the Physics Builder Currently includes several definitions of: The equivalent strain ε The damage evolution law ω κ Local version: Different regularization techniques Non-local version: One additional material parameter Both versions in the same interface Local version Non-local version
Contents Introduction Isotropic damage mechanics and localization Implementation in Comsol Multiphysics Examples Conclusions
Plain Concrete Uniaxial tension Extension of a bar by a prescribed displacement Highlight the differences of the two formulations Tensile strength of a single element (red) reduced by 5 % to force strains to localize Local model: Strains localize in one element Load-displacement curve has the same shape as the strain softening curve Non-local model: Strain localization distributed over elements Load-displacement curve influenced by the development of the localization zone
Plain Concrete Uniaxial tension
Plain Concrete Mixed mode fracture Test series by Arrea and Ingraffea (1982) Notched beam under 4-point bending to simulate a curved crack trajectory Local and Non-local model with same mesh (7.5 mm) and material parameters NL model uses quadratic interpolation NL gives better estimate of peak load but underestimates the softening Due to difference in crack trajectory? Local model Non-local model
Plain Concrete Mixed mode fracture Local model Non-local model Extent of damage
Reinforced concrete 4-point bending Application of the implemented model to a more complicated problem Only Local model Both tensile and compressive damage Heavily reinforced concrete beam tested by Leonhardt (1972) Failure due to inclined crack from support to load point Reinforcement remain elastic, included as truss elements Triangular grid to minimize the mesh bias of cracks, ~15 mm symmetry
Reinforced concrete 4-point bending First inclined crack agrees with the maximum reported load Followed by additional inclined cracks Ultimate failure due to combination of inclined cracks and chrushing
Contents Introduction Isotropic damage mechanics and localization Implementation in Comsol Multiphysics Examples Conclusions
Conclusions Implementation of an isotropic damage mechanics model to complement the solid mechanics features of Comsol Multiphysics Enables efficient analysis of cracking in quasi-brittle materials The model is introduced in a custom physics interface Two different regularization techniques are studied to ensure mesh objectivity of solutions during strain localization The model is applied to both plain and reinforced concrete with good agreement between simulated and experimental results
Thank you for your attention! Tobias Gasch KTH / Concrete Structures Vattenfall AB
References Arrea M. and Ingraffea AR. (1982). Mixed-mode crack propagation in mortar and concrete, Cornell Univeristy, Ithaca, USA Bažant ZP. and Oh B. (1983). Crack band theory for fracture of concrete, Matériaux et Construction, 5, 155-77. Cervenka V., Pukl R., Ozbolt Z. and EligenhausenR. (1990). Mesh sensitivity effects in smeared finite element analysis of concrete fracture. In: Proceedings of FraMCoS-2, 1387-96, Freiburg, Germany. Jirásek, M. (2011). Damage and Smeared Crack Models. In: Hofstetter, G. and Meschke, G (eds.), Numerical Modelling of Concrete Cracking, Springer, Wien, Austria. Leonhardt F. (1972). On the reduction of shear reinforcement as derived from the Stuttgart shear tests 1961-1963, IABSE congress report 7. Oliver J., Cervera M., Oller S., and LublinerJ. (1990). Isotropic damage models and smeared crack analysis of concrete. In: Proceedings of SCI-C 2, 945-57, Zell am See, Austria. Peerlings RHJ., De Borst R., Brekelmans WAM. and De Vree JHP. (1996). Gradient enhanced damage for quasi-brittle materials, International Journal of Numerical Methods in Engineering, 39, 3391-403.